$L_p$ space

L_p

space

Let

(X,\mathcal{A},\mu)

be a measure space and

1\leq p<\infty

. The space

L_p(X,\mu)

consists of measurable functions

f:X\to\mathbb{R}

satisfying

\begin{equation*}\|f\|_p := \left(\int_X |f(x)|^p\,\mathrm{d}\mu(x)\right)^{1/p}<\infty.\end{equation*}

Essential boundedness and

L^{\infty}

space

A measurable function

f:X\to\mathbb{R}

is essentially bounded if there is a constant

0\leq K<\infty

such that

|f(x)|\leq K

for

\mu

-almost every

x\in X

. The infimum of such bounds is called the essential supremum of

|f|

, denoted by

\begin{equation*}\|f\|_{\infty} := \operatorname{ess\,sup}_{x\in X}|f(x)| = \inf\{M\geq 0: |f(x)|\leq M \text{ for }\mu\text{-almost every }x\in X\}.\end{equation*}

Hölder and Minkowski inequalities

This section covers two most important inequality for

L_p(X, \mu)

spaces: Hölder's and Minkowski's inequalities.

Hölder inequality

Convexity of

\exp(x)

HÖLDER INEQUALITY

Suppose that

1 \leq p < \infty

and

q = \frac{p}{p-1}

. If

f \in L_p(X, \mu)

and

g \in L^q(X, \mu)

, then

fg \in L^1(X, \mu)

. Moreover,

\begin{equation*}\int_X |f(x)g(x)| \, d\mu(x) \leq \left( \int_X |f(y)|^p \, d\mu(y) \right)^{1/p} \cdot \left( \int_X |g(z)|^q \, d\mu(z) \right)^{1/q}.\end{equation*}

Proof. Let

a, b > 0

and let

A = \ln(a^p)

and

B = \ln(b^{q})

. Since the exponential function is strictly convex,

\begin{equation*}\exp \left( \frac{A}{p} + \frac{B}{q} \right) \leq \frac{\exp A}{p} + \frac{\exp B}{q} ,\end{equation*}

with equality only if

A = B

. Hence

\begin{equation*}\exp \left( \frac{\ln(a^p)}{p} + \frac{\ln(b^q)}{q} \right) = ab \leq \frac{a^p}{p} + \frac{b^{q}}{q}.\end{equation*}

The function

fg

is defined a.e. and measurable. If we set

a = |f(x)| / \|f\|_p

and

b = |g(x)| / \|g\|_q

then by previous inequality we get

\begin{equation*}\frac{|f(x)g(x)|}{\|f\|_p\|g\|_q} \leq \frac{1}{p} \frac{|f(x)|^p}{\|f\|_p^p} + \frac{1}{q} \frac{|g(x)|^q}{\|g\|_q^q}.\end{equation*}

Integrating we get

\begin{equation*}\int_X \frac{|f(x)g(x)|}{\|f\|_p\|g\|_q} \, d\mu(x)\leq\frac{1}{p} \int_X \frac{|f(x)|^p}{\|f\|_p^p} \, d\mu(x)+\frac{1}{q} \int_X \frac{|g(x)|^q}{\|g\|_q^q} \, d\mu(x)= \frac{1}{p} + \frac{1}{q} = 1.\end{equation*}

This is exactly Hölder's inequality.

\Box

An immediate corollary of Hölder’s inequality is the following Cauchy–Schwarz inequality

CAUCHY–SCHWARZ INEQUALITY

f \in L^2(\mu)

g \in L^2(\mu)

. Then

fg \in L^1(\mu)

and

\begin{equation*}\int_X f(x)g(x) \, d\mu(x) \leq \left( \int_X |f(y)|^2 \, d\mu(y) \right)^{1/2} \left( \int_X |g(z)|^2 \, d\mu(z) \right)^{1/2}.\end{equation*}

Minkowski’s inequality

To establish the Minkowski inequalities, we first introduce the following simple lemma.

Convexity of

x^4

LEMMA

1 \leq p<\infty

and

a, b \geq 0

, then

\begin{equation*}(a+b)^p \leq 2^{p-1}\left(a^p+b^p\right) .\end{equation*}

Proof. If

p=1

, then equality holds. For

p>1

, the function

t^p

is convex on

[0, \infty)

; that is, its graph lies below the chord line joining the points

\left(a, a^p\right)

and

\left(b, b^p\right)

. Thus

\begin{equation*}\left(\frac{a+b}{2}\right)^p \leq \frac{a^p+b^p}{2},\end{equation*}

from which the statement of lemma follows at once.

\Box

MINKOWSKI’S INEQUALITY

Let

1 \leq p < \infty

and let

f, g \in L_p(X, \mu)

. Then

f + g \in L_p(\mu)

and

\begin{equation*}\left( \int_X |f(x) + g(x)|^p \, \text{d} \mu(x) \right)^{1/p} \leq \left( \int_X |f(x)|^p \, \text{d} \mu(x) \right)^{1/p} + \left( \int_X |g(x)|^p \, \text{d} \mu(x) \right)^{1/p}\end{equation*}

or in the norm form

\begin{equation*}\|f + g\|_{L_p(X, \mu)} \leq \|f\|_{L_p(X, \mu)} + \|g\|_{L_p(X, \mu)} .\end{equation*}

\quad

Proof.Consider the simple inequality

\begin{equation*}|f + g|^p = |f+g| \cdot |f+g|^{p-1} \leq |f|\cdot|f+g|^{p-1} + |g|\cdot|f+g|^{p-1},\end{equation*}

and integrate over

X

, then apply Hölder's inequality to obtain

\begin{align*}&\int_X |f(x) + g(x)|^p \, \text{d} \mu(x) \leq \\&\quad \int_X |f(x)| \cdot |f(x) + g(x)|^{p-1} \, \text{d} \mu(x)+ \int_X |g(x)| \cdot |f(x) + g(x)|^{p-1} \, \text{d} \mu(x) \\&\overset{\text{Hölder's}}{\leq} \left(\|f\|_{L_p(X,\mu)} + \|g\|_{L_p(X,\mu)}\right) \cdot \left\||f+g|^{p-1}\right\|_{L^q(X,\mu)}.\end{align*}

Since

\begin{equation*}q = \frac{p}{p-1} \quad \Longrightarrow \quad \left\| | f + g |^{p-1}\right\|_{L^q(X, \mu)} = \left( \int_X | f(x) + g(x) |^{p} \text{d} \mu(x) \right)^{\frac{1}{q}}\end{equation*}

from which we obtain by dividing previous inequality by

\left( \int_X |f + g|^p \, d\mu \right)^{1/q}

\begin{equation*}\left( \int_X |f(x) + g(x)|^p \, \text{d} \mu(x) \right)^{1 - \frac{1}{q}} \leq \| f \|_{L_p(X, \mu)} + \| g \|_{L_p(X, \mu)}\overset{1-\frac{1}{q}=\frac{1}{p}}{\Longrightarrow}\|f + g\|_p \leq \|f\|_p + \|g\|_p. \qquad \Box\end{equation*}

THEOREM

For

1 \leq p \leq \infty

L_p(X, \mu)

is a normed linear space

Proof. The Minkowski's inequality means that, whenever

1 \leq p < \infty

, the formula

\begin{equation*}d(f, g) := \|f - g\|_p\end{equation*}

defines a metric on

L_p(\mu)

, since the triangle inequality holds true

\begin{equation*}\|g - h\|_p = \|g - f + f - h\|_p \leq \|g - f\|_p + \|f - h\|_p. \Box\end{equation*}

Completeness of $L_p(X,\mu)$

This is Theorem 2.16 from Adams and Fournier [AdamsFournier2003].

We want to show that the spaces

L_p(\mu)

, where

1 \leq p \leq \infty

are Banach spaces. In order to do it we need to prove that these spaces are complete, e.g. for every Cauchy sequences

\left\{f_n\right\}

\begin{equation*}\forall \varepsilon > 0, \; \exists N \text{ such that } \forall n, m \geq N: \|f_n - f_m\|_p = \left( \int_X | f_m - f_n |^p \, d\mu \right)^{\frac{1}{p}} < \varepsilon ,\end{equation*}

exist a limit function inside

L_p(\mu)

THEOREM

The space

L_p(X,\mu)

, where

1 \leq p \leq \infty

, is complete with respect to the metric

\|f - g\|_p

The Cauchy sequence

First assume

1 \leq p<\infty

and let

\left\{f_n\right\}

be a Cauchy sequence in

L_p(\mu)

. We can extract subsequence

\left\{f_{n_j}\right\}

\left\{f_n\right\}

such that

\begin{equation*}\| f_{n_{j+1}}-f_{n_j}\|_p \leq \frac{1}{2^j}, \quad j=1,2, \ldots\end{equation*}

The convergent monotone sequence

Let's define the monotonicly increasing sequence

\{ g_n\}^{\infty}_{n=1}

for

x \in X

\begin{equation*}g_m(x) = \sum_{j=1}^m | f_{n_{j+1}}(x)-f_{n_j}(x) | \quad \Longrightarrow \quad 0 \leq g_1 \leq g_2 \leq \dots \leq g_m \leq \dots\end{equation*}

and due to Minkowski's inequality for any

m

\begin{equation*}\int_X g^p_m(x) \text{d} \mu(x) \leq \left( \| f_1 \|_{L_p(X, \mu)} + \sum_{j=1}^m \| f_{n_{j+1}}-f_{n_j} \|_{L_p(X, \mu)} \right)^{p} \leq 1 ,\end{equation*}

we get that

\{ g_n\}

is bounded in

L_p(X, \mu)

. Putting

g^p(x) = \lim_{m \rightarrow \infty} g^p_m(x)

, we obtain

\begin{equation*}g_m^p(x)\uparrow g^p(x)\overset{\text{Monotone Convergence Theorem}}{\Longrightarrow}\int_X g^p(x) \text{d} \mu(x) = \lim_{m \rightarrow \infty} \int_X g^p_m(x) \text{d} \mu(x) \leq 1.\end{equation*}

Hence

g^p(x) < \infty

\mu-

a.e on

X

and

g \in L_p(X, \mu)

Pointwise convergence of $\{ f_{n_j}\}$

Our next goal is to prove that

\{f_{n_j}(x)\}

is a Cauchy sequence for almost every

x\in X

, and hence converges pointwise almost everywhere. For all

k \geq 1,

\begin{equation*}| f_{n_{m+k}}-f_{n_m} | = | f_{n_{m+k}} - f_{n_{m+k-1}} + f_{n_{m+k-1}} - \dots + f_{n_{m+1}}- f_{n_m} | \leq \sum_{i=m+1}^{k} | f_{n_{i}}-f_{n_{i-1}} | = g_{m+k} - g_m \rightarrow 0 \quad \mu-\text{a.e.}\end{equation*}

Therefore,

\{ f_{n_{m}}(x) \}_{m = 1}^{\infty}

is Cauchy sequence on real line, so it converges. Now we can define the limit

\begin{equation*}f(x)=\lim_{m \rightarrow \infty} f_{n_m} (x) \quad \mu-\text{a.e. on } X\end{equation*}

Since

|f_{n_m}| \leq g_m \leq g

\mu-

a.e,

|f| \leq g

and

|f_{n_m} - f| \leq 2^p g^p

. Then by Dominated Convergence Theorem,

\begin{equation*}\lim_{m \rightarrow \infty} \int_{X} | f - f_{n_m} |^p \text{d} \mu(x) = \int_{X} \lim_{m \rightarrow \infty} | f - f_{n_m} |^p \text{d} \mu(x) = 0\end{equation*}

Convergence to $f$

For any

\epsilon>0

there exists

N

such that if

m, n \geq N

, then

\left\|f_m-f_n\right\|_p<\epsilon

. Hence, by Fatou's lemma

\begin{align*}\int_{X}\left|f(x)-f_n(x)\right|^p \text{d} \mu(x) & =\int_{X} \lim _{j \rightarrow \infty}\left|f_{n_j}(x)-f_n(x)\right|^p \text{d} \mu(x) \\& \leq \liminf _{j \rightarrow \infty} \int_{X}\left|f_{n_j}(x)-f_n(x)\right|^p \text{d} \mu(x) \leq \epsilon^p.\end{align*}

Thus

\left\|f-f_n\right\|_p \rightarrow 0

n \rightarrow \infty

. Therefore

L_p(\Omega)

is complete and so is a Banach space.

Completeness of $L^{\infty}(X, \mu)$

Finally, if

\left\{f_n\right\}

is a Cauchy sequence in

L^{\infty}(X, \mu)

, then there exists a set

A \subset X

having measure zero such that if

x \notin A

, then for every

n, m=1,2, \ldots

\begin{equation*}\left|f_n(x)\right| \leq\left\|f_n\right\|_{\infty}, \quad\left|f_n(x)-f_m(x)\right| \leq\left\|f_n-f_m\right\|_{\infty} .\end{equation*}

Therefore,

\left\{f_n\right\}

converges uniformly on

X / A

to a bounded function

f

. Setting

f=0

for

x \in A

, we have

u \in L^{\infty}(\Omega)

and

\left\|f_n-f\right\|_{\infty} \rightarrow 0

n \rightarrow \infty

. Thus

L^{\infty}(X, \mu)

is also complete and a Banach space.

\Box

Approximation $L_p(\mathbb{R}^n)$ by Simple and Continuous Functions

This is Theorem 2.19 from Adams and Fournier [AdamsFournier2003].

Simple functions

Approximation is one of the basic tools in

L_p

theory. Many statements are first proved for very simple functions and then extended to general

L_p

functions by density. We begin with simple functions.

THEOREM

p \in [1, \infty)

, then the set of simple functions

f = \sum_{i=1}^n a_i \mathbf{1}_{E_i}

, where each

E_i

is an element of the

\sigma

-algebra

\mathcal{X}

and

\mu(E_i) < \infty

, is dense in

L_p(X, \mathcal{X}, \mu)

\quad

Proof. If

f \in L_p

, then

f

is measurable; thus, there exists a sequence

\{\phi_n\}_{n=1}^\infty

of simple functions, such that

\phi_n \rightarrow f

almost everywhere with

\begin{equation*}0 \leq |\phi_1| \leq |\phi_2| \leq \cdots \leq |f|,\end{equation*}

i.e.,

\phi_n

approximates

f

from below. For approximation sequence we get

|\phi_n - f|_p \rightarrow 0

\mu

-a.e. and

|\phi_n - f| \leq 2|f|

Since

|f|^p \in L^1(\mu)

, the Dominated Convergence Theorem gives

\begin{equation*}\|\phi_n-f\|_p^p = \int_X |\phi_n-f|^p\,d\mu \to 0. \quad \Longrightarrow \quad \|\phi_n-f\|_p \to 0. \quad \Box\end{equation*}

Continuous functions $C_0(\mathbb{R}^n)$

Simple functions are useful for measure-theoretic arguments, but in analysis it is often more convenient to work with continuous functions. On

\mathbb R^n

, every

L_p

function can in fact be approximated by continuous functions with compact support.

THEOREM

C_0(\mathbb{R}^n)

is dense in

L_p(\mathbb{R}^n)

1 \leq p<\infty

\quad

Proof. Let

u \in L_p(\mathbb{R}^n)

. As we have seen in the previous theorem, there exists a nonnegative sequence of simple functions

\{\phi_n\}

and some

\phi \in \{\phi_n\}

such that

\|u-\phi\|_p<\epsilon / 2

. Since

\phi

has compact support, it vanishes outside some bounded measurable set

E

of finite measure. By Lusin's theorem, there exists

\gamma \in C_c(\mathbb R^n)

such that

\begin{equation*}|\gamma(x)|\le \|\phi\|_\infty \quad \text{for all } x,\end{equation*}

and

\begin{equation*}\operatorname{vol}\{x:\gamma(x)\neq \phi(x)\}<\left(\frac{\varepsilon}{4\|\phi\|_\infty}\right)^p.\end{equation*}

Therefore

\begin{equation*}\|\phi-\gamma\|_p\le\|\phi-\gamma\|_\infty\,\operatorname{vol}\{x:\gamma(x)\neq \phi(x)\}^{1/p}\overset{\text{Lusin's theorem}}{<}2\|\phi\|_\infty \cdot \frac{\varepsilon}{4\|\phi\|_\infty}=\frac{\varepsilon}{2}.\end{equation*}

Hence

\begin{equation*}\|u-\gamma\|_p \le \|u-\phi\|_p + \|\phi-\gamma\|_p < \varepsilon. \quad \Box\end{equation*}

Separability of $L_p(\mathbb{R}^n)$

This is Theorem 2.21 from Adams and Fournier [AdamsFournier2003].

In this section we assume that

\Omega \subset \mathbb R^n

is open and that

\mu

is Lebesgue measure.

THEOREM

\Omega \subset \mathbb R^n

is open and

1 \le p < \infty

, then

L_p(\Omega)

is separable.

\quad

Proof. For

m=1,2, \ldots

let

\begin{equation*}\Omega_m=\{x \in \Omega:|x| \leq m \text { and } \operatorname{dist}(x, \operatorname{bdry}(\Omega)) \geq 1 / m\} .\end{equation*}

Then

\Omega_m

is a compact subset of

\Omega

. Let

P

be the set of all polynomials on

\mathbb{R}^n

having rational-complex coefficients, and let

P_m=\left\{\mathbf{1}_{\Omega_m} f: f \in P\right\}

P_m

is dense in

C\left(\Omega_m\right)

. Moreover,

\bigcup_{m=1}^{\infty} P_m

is countable. If

u \in L_p(\Omega)

and

\epsilon>0

, there exists

\phi \in C_0(\Omega)

such that

\|u-\phi\|_p<\epsilon / 2

. If

1 / m<\operatorname{dist}(\operatorname{supp}(\phi), \operatorname{bdry}(\Omega))

, then there exists

f

in the set

P_m

such that

\|\phi-f\|_{\infty}<(\epsilon / 2)\left(\operatorname{vol}\left(\Omega_m\right)\right)^{-1 / p}

. It follows that

\begin{equation*}\|\phi-f\|_p \leq\|\phi-f\|_{\infty}\left(\operatorname{vol}\left(\Omega_m\right)\right)^{1 / p}<\epsilon / 2\end{equation*}

and so

\|u-f\|_p<\epsilon

. Thus the countable set

\bigcup_{m=1}^{\infty} P_m

is dense in

L_p(\Omega)

and

L_p(\Omega)

is separable.

\Box

References

[AdamsFournier2003]Adams, Robert A., and John J. F. Fournier. Sobolev Spaces. Second edition. Academic Press, 2003.

LpL_pLp​ space

Hölder and Minkowski inequalities

Hölder inequality

Minkowski’s inequality

Completeness of Lp(X,μ)L_p(X,\mu)Lp​(X,μ)

The Cauchy sequence

The convergent monotone sequence

Pointwise convergence of {fnj}\{ f_{n_j}\}{fnj​​}

Convergence to fff

Completeness of L∞(X,μ)L^{\infty}(X, \mu)L∞(X,μ)

Approximation Lp(Rn)L_p(\mathbb{R}^n)Lp​(Rn) by Simple and Continuous Functions

Simple functions

Continuous functions C0(Rn)C_0(\mathbb{R}^n)C0​(Rn)

Separability of Lp(Rn)L_p(\mathbb{R}^n)Lp​(Rn)

References

$L_p$ space

Completeness of $L_p(X,\mu)$

Pointwise convergence of $\{ f_{n_j}\}$

Convergence to $f$

Completeness of $L^{\infty}(X, \mu)$

Approximation $L_p(\mathbb{R}^n)$ by Simple and Continuous Functions

Continuous functions $C_0(\mathbb{R}^n)$

Separability of $L_p(\mathbb{R}^n)$